Propositional Functions Research Articles

Relating Ontology and Logic

(P. 22) 172 Reviews RELATING ONTOLOGY AND LOGIC MICHAEL SCANLAN Philosophy / Oregon State U. Corvallis, OR 97331, USA SCANLANM@UCS.ORST.EDU Bernard Linsky. Russell's Metaphysical Logic. Stanford, Calif.: CSLI [Center for the Study of Language and Information] Publications, 1999; distributed by Cambridge U. P., 2000. Pp. viii, 150. Cloth: £37.50 (US$59.95); pb £13.95 (US$22·95)· The thesis of this book is that Russell's work in logic cannot be understood separately from his ontological ideas, here somewhat archaically labeled "metaphysical". The eight chapters are largely independent of each other, four of them having been published previously as journal articles. Some parts of the book make plausible the value of considering Russell's ontic commitments in evaluating his approach to issues of logic. In other places, however, the book's thesis seems to get lost. The time period examined is Russell's classic period from "On Denoting" (1905) through the logical atomism publications of1918 and 1924.1 The central work in this p~riod is clearly Principia Mathematica.. Previous writers have presented Russell in this period as constantly changing his thinking on such I Linsky dates his own period ofcentral concern in this book as 1908-19. Reviews 173 topics as the nature of propositions and the nature of logic and vieW" PM as somewhat cobbled together to show how the logicist project could be carried out, while leaving a good many loose ends. Professor Linsky will have none of this. In Chapter I he argues for his basic view that the seemingly later logical atomist ontology ofparticulars and universals which somehow combine to form facts is the Russellian ontology which is compatible with the logical framework of PM This interpretation presents a robustly realistic Russell, and not a nominalist or ontic minimalist, as Russell comes out on some others' interpretations . Linsky correctly argues against reading back, through Quine and later interpretations of "On Denoting", an interest on Russell's part in ontic economy for its own sake. In Chapter 2, Linsky addresses a crucial point for his project by considering how the ontic concept of universals is related to "propositional functions" in PM One approach in the literature is to treat propositional functions as linguistic devices which represent universals, akin to the open sentences of present-day logic. Linsky takes this interpretation to be incompatible with Russell's realist ontology and his construal of propositions as non-linguistic objects composed of actual entities. On Linskey's interpretation propositional functions are fragments ofpropositions and the propositions themselves are constructions out of particulars and universals. One problem here is showing how this squares with PM, in which propositional functions and not propositions are to be basic. It seems to me that Linsky "gives away the store" with respect to his general interpretative thesis when he says Russell himself did not have any clear idea about how to fit together his ontic views about universals with the logical framework of propositional functions in PM: When working on metaphysics, Russell would think of universals; when working on logic, his focus was on propositional functions. When combining the two, as in the introduction to PM, he did have an idea ofhow they fit together, but such mixed occasions were rare.... My proposal attributes only marginal consciousness on Russell's part to the distinction between universals and propositional functions. This rather tenuous relation between Russell's logical work and his ontic views does not bode well for a unified view of his "metaphysical logic". Chapter 3 considers a related topic, Russell's propositions, which are somehow composed of particulars and universals and are somehow related to propositional functions. Linsky is clear that up until he adopted the multiple-relation theory ofpropositions around 1908, Russell took propositions to be certain complexes of non-linguistic, non-referential entities. But with the multiplerelation theory propositions were no longer themselves entities that could occur in relations. Instead, there are certain many-place (multiple) relations contain- 174 Reviews ing a mind and a number of objects (i.e. beliefs). They are true if there is a "fact" that (somehow) corresponds to the belief, false otherwise. The critical thing for Russell is that there is...

Russellian Propositions

Reviews 181 RUSSELLIAN PROPOSITIONS GIOVANNI DE CARVALHO Rua Adriano, 102/801 cob 20735-060 Rio de Janeiro, Brazil GIOVANNIDUARTE@UOL.COM.BR Guillermo Hurtado. Proposiciones Russellianas. Mexico City: Universidad Nacional AutonoIlla de Mexico, 1998. Pp. 337. $120.00 Mexican pesos. G iven its scope and the amount of problems with which it deals, this bO.:Jk is not an easy one. Hurtado assigned himself the job ofscrutinizing an old project Russell had plit aside under severe criticism from Wittgenstein. I'm talking about the 1913 manuscript of Theory ofKnowledge. Wittgenstein showed Russell that the theory of belief he presented in that manuscript was inconsistent . Hurtado was particularly intrigued by the fact that Russell was paralysed by Wittgenstein's famous objection. The problem was a hard one, he admitted, bur it could be resolved, given enough effort. Then why didn't Russell resolve it? Hurtado's inquietude led him to proceed with an investigation, which resulted in the writing of this book. The book provides a completely new and systematic account of Russell's early philosophy by focusing on its cornerstones: Russell's ideas of predication, propositional function and the proposition. Hurtado claims to have fullyexamined what Russell said about those subjects during the years 1900 through 1913, which are considered his most productive years. It turns out that these conceptions are far more profound and obscure than is currently supposed. His task was to examine them under a new light. The book has a hybrid nature, in which historical facts are mixed together with Hurtado's own philosophical insights into Russell's ideas. In spite of Hurtado's efforts to separate things, this way of exposing Russell's viewpoint in contrast to his own, in my opinion, may bring some confusion, and at times it is hard to tell whether he is elucidating what Russell said or stating his own 182 Reviews ideas about it. Hurtado divides the book into three parts, dealing respectively with (1) relations and predication; (2) denotation and propositional functions; and (3) descriptions, logical types and beliefs. In the epilogue he proposes what he claims to be his own contribution to "a neo-Russellian theory of predication". The first part of the book deals with relations and predication. It is currently accepted that Russell vigorously rejected the idea that all relations are internal. But Hurtado shows that neither Russell nor the idealists were very clear about what an internal relation is. It is against this background that Russell's doctrines of external relations must be understood. And what about Russell's conception of predication? Here Hurtado sees Russell subject to apparently antagonistic intuitions. On pages 84-5 of The Principles ofMathematics, Russell supports the intuition that the copula is not a relation but has a relational nature. In Chapter IV of the same book he says that predication is logically a relation but is not a relation. This means that Russell recognizes the relational nature ofpredication but cannot accommodate this view in the framework of his ontology. The problem remains offinding a substratum for predication, something that relates without being a relation. In the epilogue Hurtado proposes a way of doing it. The second part of the book deals with denotation and propositional functions . Hurtado is particularly intrigued by the fact that Russell's positions regarding propositional functions have always been ambivalent. It is true, he points out, that on occasions, in the Principles, Russell held the position that we have to take the notion of propositional function as primitive, but Hurtado thinks this position must be understood in the context of the Russellian logicist project in the Principles. To the specific ends of that project (mainly to reduce mathematics to formal logic without the theory of types), yes, we have to take the notion of propositional function as primitive. On the other hand, Russell also said many interesting things about propositional functions that go far beyond the necessities of the logicist project. If, according to Hurtado, we can find in the Principles the elements for a theory of propositional functions, then why did Russell say that we have to take this notion as primitive? Hurtado's interpretation is, in his own words, that "maybe Russell...

Russell’s Modal Logic?

Reviews 165 RUSSELrs MODAL LOGIC? GARY OSTERTAG Philosophy I New York U. New York, NY 10003, USA G02@NYU.EDU Jan Dejnofka, Bertrand Russell on Modality and Logical Relevance. Aldershot, U.K.: Ashgate, 1999. Pp. ix, 241. £37.50; US$65.95 (cloth). Russell's discussions of modality hardly form a centrepiece of his philosophical logic. He published no articles on this topic,I nor do any of his books devote even a single chapter to. it. Nonetheless, remarks on modality appear-although with no great frequency-scattered throughout his writings, spanning the entire period in which he was seriously engaged in logical matters. I "Non-Euclidean Geometry" (Papers 4: 482-5), first published in 19°4, is arguably an exception. r66 Reviews In a 1905 manuscript only recently published, he argues that modal predicates are correctly applied not to propositions bur to propositional functions.2 Moreover , in a roughly coeval review of Meinong, serialized in Mind, he expresses scepticism towards the view that "it is possible to distinguish, among true propositions , some which are necessary from others which are mere facts" (Papers 4: 435). His views on modality receive further clarification in his 1913 Aristotelian Society lecture, "On the Notion of Cause"; but a more fully developed picture does not emerge until 1918-19. Two writings of that period, "The Philosophy of Logical Atomism" and Introduction to Mathematical Philosophy, state Russell's most considered thoughts on this topic. Even here, Russell does not devote an extensive amount of space to the subject. But their individual brevity notwithstanding , Russell's remarks, collectively, form a view of modality that is worth considering in detail, since it not only sheds light on his conceptions of logic and logical truth, but also provides an interesting, if incomplete, reworking of the traditional approach. Jan Dejnoika's new book attempts to provide an interpretation ofthese remarks. While Dejnofka is not the first Russell scholar to address them, his book is by far the most comprehensive discussion yet published ) Moreover, his interpretation of Russell, if correct, would require a radical reconception of the logic of Principia Mathematica. As I shall argue, Dejnoika's book falls far short ofestablishing anything so revolutionary. As early as 1905, Russell has settled into what becomes his official view, that "there is no such comparative and superlative of truth as is implied by the notions ofcontingency and necessity" (Papers 4: 520). Later, in "On the Notion of Cause", he considers the doctrine that "That is necessary which is not only true, but would be true under all circumstances."4 His response is definitive: A proposition is simply true or false, and that ends the matter: there can be no question of "circumstances." "Charles I's head was cut off' is just as true in summer as in winter, on Sundays as on Mondays. (Papers 6: 194) The notions of necessity, possibility, etc., are therefore correctly applied to propositional functions, not to propositions. In particular, a propositional function f(x) is necessary just in case it determines a true proposition for each value of x, possible just in case it determines a true proposition for at least one value of x, and impossible just in case there is no value of x for which it determines a true proposition. Brief reflection will reveal that the property of being 1 "Necessity and Possibility" (Papers 4: 5°7-20). The suggestion also occurs in Russell's review of Meinong; see Papers 4: 436, 450, . 3 Which is not to imply that there are not striking omissions; see note 8. 4 Taken from Baldwin's Dictionary ofPhilosophy and Psychology. Reviews r67 necessary, understood in the above sense, is precisely what the universal quantifier expresses, whereas the property of being possible is what existential quantification expresses. (Being impossible is expressed by the existential quantifier prefixed by the negation sign.) Most readers pause, however briefly, at these definitions..What recommends the use of "possible" as applying to a function that is, in Russell's words, "sometimes true"? One suspects that Russell is using the expression in a way that deviates from its standard use in philosophy . The suspicion is confirmed in "The Philosophy ofLogical Atomism": When I...

A puzzle about de rebus beliefs

and he found that their logic cannot be adequately formalized within the first-order predicate calculus. If we try to formalize the sentence by a paraphrase using individual variables that range over critics, or over sets or collections or fusions of critics, we misrepresent its logical structure. To represent plural quantification adequately requires the logical resources of the full second-order predicate calculus.1 The thoughts expressed by plurally quantified statements can be objects of belief, consternation, wonder, puzzlement, and desire, just as other thoughts can, but, when we try to describe the propositions such sentences express, a curious puzzle arises. There is a de dicto reading of ‘Ralph believes there are critics who admire only one another’, according to which Ralph believes the proposition that is expressed by the Geach-Kaplan sentence; that reading is unproblematic. The puzzle comes when we turn to what we may call the de rebus reading, according to which there are some critics of whom Ralph believes that they admire only one another.2 The standard theory of propositions accounts for the distinction between de dicto or notional3 propositional attitude reports and de re or relational propositional attitude reports by proposing that the two kinds of reports describe two kinds of propositions. The de dicto reading of ‘Ralph believes someone is a spy’ attributes to Ralph the belief in an existential proposition that can be symbolized ‘$x x is a spy’. The de re reading avows that Ralph believes one of the de re or singular propositions in the range of the propositional function defined by the English open sentence ‘that he is a spy’; for some individual x, Ralph believes of x that he is a spy. Analogously, we may understand the de rebus reading of ‘Ralph believes there ANALYSIS 60.4 October 2000

A puzzle about de rebus beliefs

and he found that their logic cannot be adequately formalized within the first-order predicate calculus. If we try to formalize the sentence by a paraphrase using individual variables that range over critics, or over sets or collections or fusions of critics, we misrepresent its logical structure. To represent plural quantification adequately requires the logical resources of the full second-order predicate calculus.1 The thoughts expressed by plurally quantified statements can be objects of belief, consternation, wonder, puzzlement, and desire, just as other thoughts can, but, when we try to describe the propositions such sentences express, a curious puzzle arises. There is a de dicto reading of ‘Ralph believes there are critics who admire only one another’, according to which Ralph believes the proposition that is expressed by the Geach-Kaplan sentence; that reading is unproblematic. The puzzle comes when we turn to what we may call the de rebus reading, according to which there are some critics of whom Ralph believes that they admire only one another.2 The standard theory of propositions accounts for the distinction between de dicto or notional3 propositional attitude reports and de re or relational propositional attitude reports by proposing that the two kinds of reports describe two kinds of propositions. The de dicto reading of ‘Ralph believes someone is a spy’ attributes to Ralph the belief in an existential proposition that can be symbolized ‘$x x is a spy’. The de re reading avows that Ralph believes one of the de re or singular propositions in the range of the propositional function defined by the English open sentence ‘that he is a spy’; for some individual x, Ralph believes of x that he is a spy. Analogously, we may understand the de rebus reading of ‘Ralph believes there ANALYSIS 60.4 October 2000

The Denoting Reader

Reviews 77 THE DENOTING READER NICHOLAS GRIFFIN Russell Research Centre / McMaster U. Hamilton, ON, Canada L8s 4M2 NGRIFFIN@MCMASTER.CA Gary Ostertag, ed. Definite Descriptions: a Reader. Cambridge, Mass., and London: MIT P., 1998. Pp. xii, 411. US$32.95 (cloth); US$30.00 (pb). Russell's theory of definite descriptions has been widely and continuously debated in the half century since Strawson published "On Referring".I Given the number of important papers written on the topic in that timenot to mention the even larger number of unimportant ones-it is more than a little surprising that there has been, until Ostertag's book, no substantial collection of readings on the topic. Ostertag's Definite Descriptions fills this gap admirably-though, as he admits at the outset, many important papers and some important topics have been left out. Among the latter he mentions descriptive pronouns (p. xi); I shall mention some others below. The papers can be divided into two main groups: four selections in which Russell's original theory is expounded and a group of eight papers taking sides on various current controversies over the adequacy of Russell's theory. In between are three more miscellaneous papers: Strawson's "On Referring", which couldn't well be omitted and, in any case, set several of the current controversies going; an extract from Carnap's Meaning and Necessity giving a version of Frege's conventional denotation proposal; and Karel Lambert's "A I Mind (1950); reprinted here as Chap. 6. 78 Reviews Theory of Definite Descriptions", which represents free logic.2 Lambert's paper is a decent introduction to free logic, but it was written too early to capture the important link between Strawson's paper and free logic which emerged with the development of supervaluational semantics for free logic. Sadly, there is nothing on Meinongian theories of descriptions, which have had a substantial revival since the 19705. Nor is there anything on more purely linguistic work on definite descriptions. Rereading Strawson's paper after all theseyears, one is struck first by how beautifully written it is. One is also struck by the fact that although it pointed to all sorts of fruitful developments, it does hardly anything to bring them about. An amusing case in point is the "very special and odd sense of 'imply'" (p. 145) in which, Strawson wants to say, "The king of France is wise" implies that there is a king of France. Strawson repeatedly invokes this sense and toward the end of his paper even refers to the sense as "by now familiar" (p. 157). But repeatedly calling something "strange" does not make it familiar, and Strawson says not one thing about it-except that it does not mean either "entail" or "assert". It is, of course, the useful notion of presupposition , given formal articulation by van Fraassen,3 against every warning by Strawson who maintained that "ordinary language has no exact logic" (p. 159). Here, and elsewhere, Strawson's paper proved seminal in spite of his best efforts. The four expository selections include three by Russell: "On Denoting" (of course), the much more accessible chapter on descriptions from Introduction to Mathematical Philosophy,4 .and the relevant sections of Principia Mathematica (pp. 30-2 and 66-71 from the Introduction, and the Summary of *14). These are well-nigh essential, but the dotty notation of Principia does pose problems for using the text in the classroom. Perhaps because I had assumed that the students would get the hang of it relatively quickly, I found myself continually obliged to stop and translate into a notation that used more brackets and fewer dots. Fortunately, Ostertag himself gives a lucid summary in more modern notation in his Introduction.5 The fourth paper in 1 From his Philosophical Applications of Free Logic (Dordrecht: Kluwer, 1962; New York: Oxford U. P., 1991)-wrongly cited in the bibliography as "Philosophical Foundations of Free Logic". 3 Cf "Presuppositions, Supervaluations, and Free Logic", in K. Lambert, ed., The Logical ~y of Doing Things (New Haven, Conn.: Yale U. P., 1969); and Formal Semantics and Logic (London: Macmillan, 1971), pp. 153-63. 4 Reprinted complete with typos: "a propositional fUnction x' (towards the foot...

Łukasiewicz Operations in Fuzzy Set and Many-Valued Representations of Quantum Logics

It, is shown that Birkhoff –von Neumann quantum logic (i.e., an orthomodular lattice or poset) possessing an ordering set of probability measures S can be isomorphically represented as a family of fuzzy subsets of S or, equivalently, as a family of propositional functions with arguments ranging over S and belonging to the domain of infinite-valued Łukasiewicz logic. This representation endows BvN quantum logic with a new pair of partially defined binary operations, different from the order-theoretic ones: Łukasiewicz intersection and union of fuzzy sets in the first case and Łukasiewicz conjunction and disjunction in the second. Relations between old and new operations are studied and it is shown that although they coincide whenever new operations are defined, they are not identical in general. The hypothesis that quantum-logical conjunction and disjunction should be represented by Łukasiewicz operations, not by order-theoretic join and meet is formulated and some of its possible consequences are considered.

Approximate real-time decision making: Concepts and rough fuzzy Petri net models

This paper considers the construction of Petri nets to simulate the computation performed by decision systems. Algorithms are given to construct Petri nets which correspond to decision rules, information systems, and real-time decision systems. Rough as well as rough fuzzy Petri net extensions of colored and generalized fuzzy Petri nets are used to create highly parallel programs to simulate reasoning system computations. Constructed nets make it possible to evaluate the design of decision system tables, and to trace computations in rules derived from decision tables. Start places of nets are connected to Dill process receptors which await input from the environment. Time consumption during the propagation of outputs from sensors in a decision system is monitored with timers called approximate time windows, which measure durations between firings of decision transitions relative to time granules with names such as early, ontime, and late. Guards on decision transitions are propositional functions which permit a rule to fire for some sensor values and not for others. In addition, the design of guards makes allowance for multivalued logic, where conditional sensor readings are assessed in terms of their degree of membership in sensor measurement granules. In some cases, a rule can fire if the degree of truth of its guard (premise) is above some threshold. Through simulation, designers can arrive at reasonable estimates of the period of timers on decision transitions. The approach to simulating computations by decision systems presented in this paper results in fast, massively parallel programs implementable on a multiprocessor. © 1999 John Wiley & Sons, Inc.

The Ramsey Test and Conditional Semantics

Proponents of the projection strategy take an epistemic rule for the evaluation of English conditionals, the Ramsey test, as clue to the truth-conditional semantics of conditionals. They also construe English conditionals as stronger than the material conditional. Given plausible assumptions, however, the Ramsey test induces the semantics of the material conditional. The alleged link between Ramsey test and truth conditions stronger than those of the material conditional can be saved by construing conditionals as ternary, rather than binary, propositional functions with a hidden contextual parameter. But such a ternary construal raises problems of its own.

On the puzzling ‘puzzled can’ in wh-questions

The author deals with such occurrences of can in wh-questions (cf. Who can have died?) where the relevant answer is identical with that to a corresponding question without can and where the conjunction of a short answer of that type and the negation of the respective verbal complement with the same referent is self-contradictory (cf. ∗ Peter, but Peter has not died). It is the author's claim that this unusual kind of occurrence of can is due, not to any lexical polysemy of the verb, but to a special scope of can. In the occurrences under consideration, can does not form part of the propositional function presented in the question, but is applied, with its ordinary force, to the concept of a definite description equivalent to the definite description of an argument of the propositional function presented in the question, the argument being suggested by the material of the question and the former definite description being something yet to be found, in accordance with the nature of questions. This interpretation is also meant to explain the special ‘puzzlement’ effect of the questions with can. The kind of definite description which is looked for is suggested also by pragmatic factors. Here, the mechanism of Gricean ‘generalised conversational implicature’ seems to be at work. The author submits his own, more detailed account of the Gricean concept and exploits it in the analysis. The whole is viewed as an illustration of the interface between semantics, syntax and pragmatics.

A Reflection on Russell's Ramified Types and Kripke's Hierarchy of Truths

Both in Kripke's Theory of Truth KTT [8] and Russell's Ramified Type Theory RTT [16, 9] we are confronted with some hierarchy. In RTT, we have a double hierarchy of orders and types. That is, the class of propositions is divided into different orders where a propositional function can only depend on objects of lower orders and types. Kripke on the other hand, has a ladder of languages where the truth of a proposition in language Ln can only be made in Lm where m)n. Kripke finds a fixed point for his hierarchy (something Russell does not attempt to do). We investigate in this paper the similarities of both hierarchies: At level n of KTT the truth or falsehood of all order-n-propositions of RTT can be established. Moreover, there are order-n-propositions that get a truth value at an earlier stage in KTT. Furthermore, we show that wrr is more restrictive than KTT, as some type restrictions are not needed in KTT and more formula., can be expressed in the latter. Looking back at the double hierarchy of Russell, Ramsey [11] and Hilbert and Ackermann [7] considered the orders to cause the restrictiveness, and therefore removed them. This removal resulted in Church's Simple Type Theory STT [1] We show however that orders in RTT correspond to levels of truth in KTT. Hence, KTT can be regarded as the dual of STT where types have been removed and orders are maintained. As RTT is more restrictive than KTT, we can conclude that it is the combination of types and orders that was the restrictive factor in RTT.

Decidability in Intuitionistic Type Theory is Functionally Decidable

AbstractIn this paper we show that the usual intuitionistic characterization of the decidability of the propositional function B(x) prop [x : A], i. e. to require that the predicate (∀x ∈ A) (B(x) ∨ ¬ B(x)) is provable, is equivalent, when working within the framework of Martin‐Löf's Intuitionistic Type Theory, to require that there exists a decision function ψ: A → Boole such that (∀x ∈ A) ((ψ(x) = Boole true) ↔ B(x)). Since we will also show that the proposition x = Boole true [x: Boole] is decidable, we can alternatively say that the main result of this paper is a proof that the decidability of the predicate B(x) prop [x : A] can be effectively reduced by a function ψ A → Boole to the decidability of the predicate ψ(x) = Boole true [x : A]. All the proofs are carried out within the Intuitionistic Type Theory and hence the decision function ψ, together with a proof of its correctness, is effectively constructed as a function of the proof of (∀x ∈ A)(B(x) ∨ ¬ B(x)).Mathematics Subject Classification: 03B15, 03B20.

The functional role of verbs in terminological definitions

The role of verbs in definitions is very complex. In general discourse, the function of declarative sentences or propositions is to report events happening to participants, with verbs encoding the events and nouns the participants. In definitions of concepts, verbs and nouns are used to express, in the form of simple (stative) propositions, basic facts concerning processes, states, or actions that relate to the concepts mainly realised in noun form. While the nouns in the definienda can represent any class of concept, in this article it is argued that the verbs occurring in the second part of the definition, in finite form in the relative clause, can only represent activity or relation concepts because they serve to express states or conditions of existence, actions or executed processes.

An interpretation of Peano's logic

The problem of an adequate interpretation of G. Peano's logic still remains open, largely due to its striking pecularities: (i) the use of the same notation to designate relations both between classes and propositions; (ii) the lack of precision in his understanding of propositional functions as distinct from propositions; (iii) the systematic use of conditional definitions.

Charles S. Peirce and the philosophy of science: papers from the Harvard Sesquicentennial Congress

Interest in Charles Sanders Peirce (1839-1914) is today worldwide. Ernest Nagel of Columbia University wrote in 1959 that is a fair consensus among historians of ideas that Charles Sanders Peirce remains most original, versatile, and comprehensive philosophical mind this country has yet produced. breadth of topics discussed in present volume suggests that this is as true today as it was in 1959. Papers concerning Peirce's philosophy of science were given at Harvard Congress representatives from Italy, France, Sweden, Finland, Korea, India, Denmark, Greece, Brazil, Belgium, Spain, Germany, and United States. Charles S. Peirce Sesquicentennial International Congress opened at Harvard University on September 5, 1989, and concluded on 10th Peirce's birthday. Congress was host to approximately 450 scholars from 26 different nations. present volume is a compilation of selected papers presented at that Congress. philosophy of science and its logic are themes in work of Charles Peirce that have been of greatest interest to scholars. Peirce was himself a physical scientist. He worked as an assistant at Harvard Astronomical Observatory from 1869 to 1872 and made a series of astronomical observations there from 1872 to 1875. Solon I. Bailey says of these observations, The first attempt at Harvard Observatory to determine form of Milky Way, or galactic system, was made Charles S. Peirce....The investigation was of a pioneer nature, founded on scant data. Peirce also made major contributions in fields as diverse as mathematical logic and psychology. C. I. Lewis has remarked that the head and font of mathematical logic are found in calculus of propositional functions as developed Peirce and Peirce subsequently invented, almost from whole cloth, semiotics - science of meaning of signs. Ogden and Richards, British critics, say that by far most elaborate and determined attempt to give an account of signs and their meanings is that of American logician C. S. Peirce, from whom William James took idea and term Pragmatism, and whose Algebra of Dyadic Relations was developed Schroeder.

Two Dogmas in Retrospect

In retrospecting ‘Two Dogmas’ I find myself overshooting the mark by twenty years. I think back to college days, 61 years ago. I majored in mathematics and was doing my honors reading in mathematical logic, a subject that had not yet penetrated the Oberlin curriculum. My new love, in the platonic sense, was Whitehead and Russell’s Principia Mathematica.I was taken with the clear, clean incisiveness of its formulas. But this was not true of its long introduction to volume 1, nor of some of the explanatory patches of prose that were interspersed through the three volumes. In those pages and passages the distinction between sign and object, or use and mention, was badly blurred. Partly in consequence, there was vague recourse to intensional properties, or ideas, under the disarmingly technical name of propositional functions. These ill-conceived mentalistic notions paraded as the philosophical foundation for the clean-cut classes, truth functions, and quantification that would have been a far better starting point in their own right.

Adams on Actualism and Presentism

In Time and Thisness .(Midwest Studies in Philosophy i i (i986): 3 I 5-z9), Robert Adams argues for what I shall call Temporal Dependency Thesis (TDT). According to TDT, no propositions about an individual and no thisnesses of exist prior to existence of individual in question, where a thisness property of being x, or of being identical with and a singular proposition about an individual x is a proposition that involves or refers to x directly, perhaps by having x or thisness of x as a constituent, and not merely by way of x's qualitative properties or relations to other individuals (p. 3 I 5). The support for this position comes from two sources, according to Adams: actualism about possible worlds (Adams says that actualism is view that the two-place propositional function rx exists in wl, where w is a possible world, is not primitive but is understood as meaning that . . . rx exists] is included in world-story of w, or that x would have existed if w had been actual (p. 3 zi)), and intuitions to effect that future is metaphysically open in a way in which past is not. I wish to argue two points here: Adams argument does not establish TDT, and his position is not grounded in our intuitions concerning openness of future. I turn first to argument for TDT. The argument for TDT involves premises which Adams claims are implications of actualism, crucial premise being claim that there are no propositions about a thing in worlds in which that thing does not exist. He says, My thisness, and propositions about me, cannot have pre-existed me because if they had, it would have been possible for them to have existed even if I had never existed, and that is not possible. (p. 3I7) Thus, if actualism is true and has implications attributed to it, TDT follows since, according to Adams, its denial is incompatible with purported implications of actualism.

Berkeley's Doctrine of Notions: A Reconstruction Based on His Theory of Meaning (review)

622 JOURNAL OF THE HISTORY OF PHILOSOPHY 27:4 OCTOBER 1989 resented Hobbes's position and makes it more difficult, in consequence, for the reader to get an accurate picture of how Hobbes would have set about answering his Lockean critics. (2) Many of Hampton's interesting and perceptive observations get lost or obscured in a text which is, I believe, much too long. This would be a better book if it were shortened or compressed. Futhermore, throughout this work Hampton employs the techniques and jargon of modern game theory. Despite Hampton's remarks (137), I am not convinced that this technical apparatus is strictly necessary or helpful. Its use could certainly have been curtailed. Finally, in order to keep the number of pages down the publisher has used a typeface which is much too small and makes reading a work of this length and complexity more taxing than it should be. Hampton's study cannot be recommended as a helpful guide for students who are fresh to Hobbes's political thought. (An exception to this is her first chapter, which is an especially interesting discussion of the psychological and ethical foundations of Hobbes's argument.) Nevertheless, as should be clear from what I have already said, it is essential reading for all those who are seriously concerned to come to grips with Hobbes's social contract theory. Hobbes scholar and political theorist alike will learn a great deal from it. PAtJL RUSSELL University of British Columbia Daniel E. Flage. Berkeley's Doctrine of Notions: A Reconstruction Based on His Theory of Meaning. New York: St Martin's Press, 1987. Pp. 226. $29.95. For the reviewer of any hermeneutic work about a classical philosopher the first, and the last, questions must be: "How much, and how, does it improve our understanding of the mind of the master?" In the present case the answers are, sadly, "Not much" and "Mainly by emphasizing the importance of Berkeley's distinction between positive and relative notions." Dr. Flage starts by explaining that he will "be concerned with the nature of notional knowledge, the ontological analysis of notions, and the relations between notions and ideas in Berkeley's metaphysics" (1). Yet, surely, for Berkeley himself only the third of these ever constituted a central concern. Even there Flage is at small pains to reveal what for Berkeley rather than for Flage was the main problem. Chapter 1, "Abstraction," contends that Locke had many predecessors as an abstractionist . But since we are never told by what criteria this creature is to be identified, and since Berkeley famously allows that there are senses in which we can all truly be said to be able to abstract, it remains unclear how many of those predecessors are involved with Locke in a common catastrophe. Chapter 2, "Possibility and Impossibility," must appall those sharing Berkeley's own commitment to speaking with the vulgar. Is it really necessary, or even possible, to elucidate his luminously expressed thoughts through pages from which the tortured reader has continually to refer to an appendix listing no less than fifty-five different propositional functions? BOOK REVIEWS 623 In his "Introduction" Flage argues, sensibly enough, that, if we are to find satisfactory answers to questions about Berkeley's doctrine of notions, "it is only reasonable to begin with an examination of his later works" (3). But then in chapter 3, "Berkeley's Theory of Meaning," Flage, almost unbelievably, contrives to make no mention whatsoever of Alciphron 7. 14. That is the section which sums tip what, "upon the whole, may be said of all signs:--that they do not always suggest ideas signified to the mind: that when they suggest ideas, they are not general abstract ideas; that they have other uses besides barely standing for and exhibiting ideas ... for instance, the algebraic mark, which denotes the root of a negative square, hath its use in logistic operations, although it be impossible to form an idea of any such quantity." I shall not here repeat my argument for construing this example as a Wittgensteinian anticipation.' But it does still have to be said that the whole section must make it difficult to maintain, without qualification...

Propositional functions and families of types.

When specifying the task of a computer program, it is often natural to use recursion on a data type. In Martin-Lof's type theory, a universe must be used when defining a propositional function by recursion. Using a logical framework for type theory, formulated by Martin-Lof, an extension of type theory is proposed by which propositional functions can be directly defined without using a universe

On the identification of properties and propositional functions

The theory of properties, relations, and propositions (PRPs) promises to play a significant foundational role in logic, philosophy, semantics, and psychology. There is now a growing recognition of this fact by research ers in these disciplines.' Nevertheless, formalizations of PRP theory run into some technical complexities in connection with the treatment of free and bound terms occurring in PRP-abstracts (see, e.g., Bealer, 1979, 1982, 1983; Zalta, 1980; Monnich, 1983). These complexities can be largely avoided by adopting the thesis that properties and propositional functions are identical (see, e.g., Aczel, 1980 and 1987; M6nnich, 1983; Turner, 1987; Chierchia and Turner, 1988). The purpose of this paper is to present some reasons against this thesis despite its short-term technical advantages.2 Philosophically, the propositional-function thesis seems unacceptable on at least two counts. First, it is highly counterintuitive. How im plausible that familiar sensible properties are functions the color of this ink, the aroma of coffee, the shape of your hand, the special painfulness of a burn or itchiness of a mosquito bite. No function is a color, a smell, a shape, or a feeling. Or consider the fact that navy blue is darker than yellow and that it covers the surface of my pen. No function is darker than another function, and no function covers the surface of my pen. To assert otherwise seems to be a category mistake.3 In addition to intuitive objections to the propositional-function thesis, a second philosophical objection is that the thesis threatens the prospect of certain explanations in epistemology, phenomenology, and philo sophical psychology. For example, how are we to explain what is going on in the interplay of sensation and cognition when a person sees that two objects have some sensible quality in common? Or how are we to explain why various shades of color can look so similar? At best, the identification of properties and propositional functions complicates our epistemology, phenomenology, and philosophical psychology in con nection with these and kindred phenomena. My aim, however, is not to dwell on such philosophical objections to the propositional-function thesis, serious as they are. Rather, my aim is to discuss four logical difficulties facing the thesis. (1) The first problem derives from the fact that functions are usually